学术报告（韩斌 6.8）

摘要: Directional representation systems are required to effectively capture edge singularities for many high-dimensional problems. In this talk, we first discuss directional tensor product complex tight framelets for image processing. However, constructing compactly supported multivariate tight framelets is known to be a challenging problem because it is linked to sum of squares and factorization of multivariate Laurent polynomials in algebraic geometry. To circumvent this difficulty, next we introduce the notion of quasi-tight framelets, which behaves almost identical to a tight framelet. From an arbitrary compactly supported multivariate refinable function (such as refinable box splines) with a general dilation matrix, we constructively prove that we can always derive a directional compactly supported quasi-tight framelet with vanishing moments. The key ingredient in constructing multivariate quasi-tight framelets is the generalized matrix spectral factorization of multivariate polynomials. This generalizes the famous one-dimensional matrix spectral factorization theorem which is known to fail in high dimensions. This talk is based on several joint works with Chenzhe Diao, Ran Lu, Zhenpeng Zhao and Xiaosheng Zhuang.